3.198 \(\int \cosh (4 x) \sinh (x) \, dx\)

Optimal. Leaf size=17 \[ \frac {1}{10} \cosh (5 x)-\frac {1}{6} \cosh (3 x) \]

[Out]

-1/6*cosh(3*x)+1/10*cosh(5*x)

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {4284} \[ \frac {1}{10} \cosh (5 x)-\frac {1}{6} \cosh (3 x) \]

Antiderivative was successfully verified.

[In]

Int[Cosh[4*x]*Sinh[x],x]

[Out]

-Cosh[3*x]/6 + Cosh[5*x]/10

Rule 4284

Int[cos[(c_.) + (d_.)*(x_)]*sin[(a_.) + (b_.)*(x_)], x_Symbol] :> -Simp[Cos[a - c + (b - d)*x]/(2*(b - d)), x]
- Simp[Cos[a + c + (b + d)*x]/(2*(b + d)), x] /; FreeQ[{a, b, c, d}, x] && NeQ[b^2 - d^2, 0]

Rubi steps

\begin {align*} \int \cosh (4 x) \sinh (x) \, dx &=-\frac {1}{6} \cosh (3 x)+\frac {1}{10} \cosh (5 x)\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 17, normalized size = 1.00 \[ \frac {1}{10} \cosh (5 x)-\frac {1}{6} \cosh (3 x) \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[4*x]*Sinh[x],x]

[Out]

-1/6*Cosh[3*x] + Cosh[5*x]/10

________________________________________________________________________________________

fricas [B]  time = 0.41, size = 38, normalized size = 2.24 \[ \frac {1}{10} \, \cosh \relax (x)^{5} + \frac {1}{2} \, \cosh \relax (x) \sinh \relax (x)^{4} - \frac {1}{6} \, \cosh \relax (x)^{3} + \frac {1}{2} \, {\left (2 \, \cosh \relax (x)^{3} - \cosh \relax (x)\right )} \sinh \relax (x)^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(4*x)*sinh(x),x, algorithm="fricas")

[Out]

1/10*cosh(x)^5 + 1/2*cosh(x)*sinh(x)^4 - 1/6*cosh(x)^3 + 1/2*(2*cosh(x)^3 - cosh(x))*sinh(x)^2

________________________________________________________________________________________

giac [B]  time = 0.11, size = 27, normalized size = 1.59 \[ -\frac {1}{60} \, {\left (5 \, e^{\left (2 \, x\right )} - 3\right )} e^{\left (-5 \, x\right )} + \frac {1}{20} \, e^{\left (5 \, x\right )} - \frac {1}{12} \, e^{\left (3 \, x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(4*x)*sinh(x),x, algorithm="giac")

[Out]

-1/60*(5*e^(2*x) - 3)*e^(-5*x) + 1/20*e^(5*x) - 1/12*e^(3*x)

________________________________________________________________________________________

maple [A]  time = 0.20, size = 14, normalized size = 0.82 \[ -\frac {\cosh \left (3 x \right )}{6}+\frac {\cosh \left (5 x \right )}{10} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(4*x)*sinh(x),x)

[Out]

-1/6*cosh(3*x)+1/10*cosh(5*x)

________________________________________________________________________________________

maxima [B]  time = 0.33, size = 27, normalized size = 1.59 \[ -\frac {1}{60} \, {\left (5 \, e^{\left (-2 \, x\right )} - 3\right )} e^{\left (5 \, x\right )} - \frac {1}{12} \, e^{\left (-3 \, x\right )} + \frac {1}{20} \, e^{\left (-5 \, x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(4*x)*sinh(x),x, algorithm="maxima")

[Out]

-1/60*(5*e^(-2*x) - 3)*e^(5*x) - 1/12*e^(-3*x) + 1/20*e^(-5*x)

________________________________________________________________________________________

mupad [B]  time = 1.43, size = 15, normalized size = 0.88 \[ \frac {8\,{\mathrm {cosh}\relax (x)}^5}{5}-\frac {8\,{\mathrm {cosh}\relax (x)}^3}{3}+\mathrm {cosh}\relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(4*x)*sinh(x),x)

[Out]

cosh(x) - (8*cosh(x)^3)/3 + (8*cosh(x)^5)/5

________________________________________________________________________________________

sympy [A]  time = 0.41, size = 20, normalized size = 1.18 \[ \frac {4 \sinh {\relax (x )} \sinh {\left (4 x \right )}}{15} - \frac {\cosh {\relax (x )} \cosh {\left (4 x \right )}}{15} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(4*x)*sinh(x),x)

[Out]

4*sinh(x)*sinh(4*x)/15 - cosh(x)*cosh(4*x)/15

________________________________________________________________________________________